I have a cartridge collection, and often tumble old tarnished rounds to clean them up. I wouldn't tumble rimfire ammo, but with centerfires I don't worry about it. If one was to go off, inside the tumbler barrel, what would happen? Not much. The brass case would rupture, and the expanding gases would raise pressure inside the drum for a second, until it bled out the seals. That's it. No big deal.
As for tumbling ammo and grinding the powder into smaller bits, thus changing the burning rate..........I'm not convinced. We used to keep extra ammo in the trunks of our patrol cars, some of it for a couple of years, and when we shot it up there was NO apparent difference. A chronograph might have showed a change, but it fired just fine, and shot to the same point of impact.
Ten years ago I loaded a bunch of ammo (two lots) for off-duty and backup use, and one lot of it wound up sitting on my floorboards for about five years, I just forgot it was there. When I got a chronograph I compared the car ammo to the other half of the same batch, and there was NO discernable difference. I pull out five rounds once a year and re-test it, and there has been no change, despite the fact that the car ammo is bounced all over creation on bumpy roads and subjected to temperature ranges that would break a rock.
The ammo in question was loaded with WST, a small-flake shotgun powder, so maybe other powder types would be affected more. But ball powder can't change much, and I can't see high-density rifle ammo having its powder ground into corn meal, either. It sounds good in theory, but I have yet to see anything close to proof that such a change actually occurs.
I'd be more inclined to believe that high temperatures would have a detrimental effect on the ammo, but that doesn't seem to be the case either.